Q  A 
300 


IN 


Answers  to  Exercises 


IN 


AN  INTRODUCTION  TO 

MATHEMATICAL 
ANALYSIS 

BY 
FRANK  LOXLEY  GRIFFIN 

PROFESSOR  OP  MATHEMATICS,  REED  COLLEGE,  PORTLAND,  OREGON 


HOUGHTON  MIFFLIN  COMPANY 

Boston        New  York        Chicago       San  Francisco 


COPYRIGHT,  1922,  BY  FRANK  LOXLEY  GRIFFIN 
ALL  RIGHTS  RESERVED 


CAMBRIDGE   •     MASSACHUSETTS 
PRINTED  IN  THE  U.  S.  A. 


QA300 


ANSWERS  TO  EXERCISES 
N.B.  Very  slight  inaccuracies  in  graphical  results  are  to  be  ignored. 

Pages  8-10 

I.   37.2  gm.,  15.3  gm.,  -6.5°.  2.  13.6  in.,  10,900  ft.,  6.8  in. 

3.  81,800,  64.7  yr.,  9600.                        5.  12.7  kg.,  3.17  lb. 
6.   $24.                                                      7.  12.3  mi. 

Pages  12-13 
I.   $3990,  23.8  yr.  2.   13.5  yr.,  49.7  yr.  3.   $680  per  yr. 

4.  -2.4  min.,  April  20,  etc.  5.    1.8  larger.  6.  23  d.,  81  d. 

Pages  16-18 

I.  <  =  1.39,  300  per  hr.,  820  per  hr.  "   2.   46.3°,  5.7  deg./min. 

3.  825  mg.,  T  =  1800, . . . ,  4-   5200  ft.,  1040  ft./min.,  610 

.26  mg./yr.  (deer.).  ft./min. 

5.  21.5mi./hr.,  . .  .,2.4mi./hr.  per       6.   .86  sq.  cm./day. 

ton/hr. 

Pages  20-21 

2.  11°.     3-   75615.  4.  25.87  in.  5.   11.72  yr. 

7.  101.43,839.2.  8.   13.5296,  28.8.         10.  .7052. 

II.  1.03,  .00075  H.P./R.P.M.,  y  =  1950  gives  p  =  1.17. 

Pages  22-24 

4.  76.56,9.3541;...  5-    ...,49046.  7.10:30. 

8.  $6550, 132  ft. ;  $12.80  per  ft.        9-    . . . ,  33  ft./min.  gained  per  min. 

10.  39600  ft. 

,  Pages  27-29 

I.   343000  ft.,  36  ft. /min.  lost  per  min.    2.    193  ft. /min.,  10.1ft. 

3.  78400  ft.  4.   502.5  ft. 

5.  3500  million  mi.,  14.5  million  mi./yr.  lost  per  yr. 
7.    171  ton-sec,  23  tons/sec.  8.    16  in.-lb. 

9.  38900  ft.-lb.  10.   12900  ft.-lb. 

11.  14500  amp. /sec,  .34  coulombs.       12.  2570  kw.-hrs. 


m55;5509 


ii  MATHEMATICAL   ANALYSIS 

Pages  30-31 

I.  2730  sq.  yd.  2.   20700  sq.  ft.  3.   223000  cu.  ft. 
4.    1,640,000  cu.  ft.;  deer.  2370  sq.ft. /ft.  5.   320  cc. 

6.   416,000  cu.  ft. 

Pages  33-35 

2.   6.25  in.                       3.   3.33  in.,  592.6  cu.  in.  4.  7.75  yd.,  15.49  yd. 

5.   3:12  P.M.,  48  mi.      6.   8.66  in.  7.  11.55  in.,  16.33in. 

8.   8.32  in.  X  12.49  in.    9.    14  X  14  X  28  cu.  in.  10.  20  yd.,  30  yd., 

$240. 

II.  6.21  ft.  X  3.11  ft.     12.   4.54 yd.  X  3.63  yd.  13.  3.92  in.,  1056 

cu.  in. 
Pages  39-40 

9='=  V57  10  5=^V"^.  1        1  J — 

'•  4       V       3'        '  12         '       "5' "5'        -2=^3 V-1; 

-3±6V3  _       ^  ,,   / ;=      1 


■;     =*=2,  =tl;     =ty2V-5±Vii3;    ^(-7=*=  V49-12c); 

l(-b±V62_j-i4o). 

2.   3.24  in.;  9.26  in.  3.   .658  in.  4.    10=fcV20. 

5.    -2.21,  .54, 1.68;   -2.19,3.19;   -2.67;  -3.97, -.49,  .52,2.95. 
7.   2.38.  8.   2.82,  12.5,  22.18.      11.   6,  only. 

Pages  41-42 

1.  -1.68,  1.61,  4.07;        -2.95,  .78,  2.17;        -1.85,  -.70,  .58,  3.97; 
3.18  only;  -4.03,  5.02  only;        -2.04  only; 

a;2  =  -3.56,  .56;  T'  -  -3.97,  .90,  3.07. 

2.  10.17  ft.  3.   4.11  in.  4.   No  imaginaries. 

Pages  44-45 
4.   96.32  ft. /sec.  5.   196  ft.,  V2(7±  V29).         6.   47.84  ft. /sec. 

7.    .244  ft.  per  ft.  9.   15800  cu.  ft.  10.   25.3,  .  .  . ,  35.8. 

Pages  49-50 
4.   Four  break,  one  twice.       5.    -25,37.5.  7.    11.86,9.65. 

^.   E  =  .08  W)  2.8.  ID.    V  =  4500/p.  11.   R  =  .072  V\ 

12.    A  =  512,000,000/i22. 


Page  53 
1.   TF  =  54+.5  T,  3.   F  =  182  +  %  T. 

5.   L  =  605  -.69  T.  '   6.  M  =  4.19&-132. 


ANSWERS  TO  EXERCISES  iii 

Pages  55-57 
3.   deer.  .84  sq.  cm. /day.  4.   $365,    $52peryr. 

6.    1835  yd.,  .062  mils/yd.  7.   29200  sq.  ft. 

8.    19400°.  9.    1380  deg./min.  gained  per  min. 

10.  4.70,  -20  per  sec,  2.54  in.  12.    -5.34;   -2.91,  -.68,  .64,  3.95. 
15.   44f^.           16.    16.2  cal,  3.33  ft.  from  A.  17.   8.91  in.,  28  in. 

Pages  62-63 
_3.    140  ft./sec.  4.   40  ft./sec.  5.   320  ft./sec. 

6.    10  sq.  in. /in.  7.    12  x  sq.  in./in.         8.    —500/441  units. 

9.   180  units.  10.    —500  units. 

Page  65 

2.  18.85  in.,  28.27  sq.  in.  3.    1100  mi.,  2100  mi. 

Pages  68-69 
I.   .75,  etc.  2.   16%,  etc.        4.   .24.  5.   2. 

6.  4.  8.   .3.  9.    -¥9;  av.  slope  =  -1/(9+3  Ax). 

Pages  73-74 
5.    100,  etc.  7.    . .  . ,  -18.75;  .  . . ,  32+27  Aa:+9  Aa:2+Ax'. 

8.   12,  . . .  9.   2.  10.    -2.5  dynes/cm. 

11.  2  ft./sec.  gained  per  sec. 

Page  77 

I.    360  -  32  <  ft./sec,  ..  .,2025  ft.  2.   3x2-6,±V2. 

3.  ^600/p2  cu.  in.  per  Ib./sq.  in.  6.   4  Trr^  cu.  ft.  per  ft. 

Page  SI 

1.  2  X,  3  a:2,  4  a^J,  .  .  . ,  2  x-9,  -7/x2,  70-10  t,  etc 

2.  .  .  . ,  245  ft.  3.    -10.25,  X  =  4.5. 
5.   4 —.4  i  ft./sec  gained  per  sec.           6.    —.6. 

7.  180  units  per  in.  8.    —  40/x^  dynes  per  cm. 

Page  84 

3.  48  ft.  per  ft./sec.  5.   1200  cu.  in. /in.        6.   3.68  ft.  per  mi./hr. 

Pages  86-87 
I.    . .  . ,  14  x^+Vs  x\  2.    .  .  . ,  10  ax5+8  3^/h,  etc 

3.  360x-60x2,     5a^+8x3+15x2-2x-2,     36x3-60x. 

4.  2.4  ft./sec.  per  sec  5.   .028. 


iv  MATHEMATICAL  ANALYSIS 

6.  15000  cu.  ft.  per  ft.  7.   3000  units/min. ;  incr.,  deer. 
8.   18;  incr.                                '            12.    -.6819,  -1.0899;  etc. 

13.   2a:+5-3M  14.   2x-22/x3. 

16.  A  lb.,  approx. 

Page  89 

3.  78.54%.  3.   52.36%. 

Pages  90-92 
I.   7;  7.0035.         2.   .215.         3.   .00093.         4.   .6  cu.  in.,  .24  sq.  in. 
5.    .0015  in.,  .0025  in.  6.    .85  sq.  in.,  .126  ft., 

7.  .00199  ft., 8.   7.54  sq.  in.,  .  .  . ,  3.77  sq.  in.  9.    .0125  cm.,  . . . 

10.   .6%,  .0125%.  II.   35.2  ft.  12.    .0123  ton,  ... . 

13.   .0138.  14.   7.2^  per  hr.,  .... 

Pages  95-96 

1.  X  =  3,  y  =  —4;  x  =  —l,y=  —1;  max.  y  =  17,  min.  y  =  —15; 
none;  max.  y  =  10,  min.  y  =  —6  (twice);  min.  y  =  —332. 

2.  .04.  3.   538.2+.  4.   16000/2^  cu.  in. 
5.   t  =  80/3^                         6.  X  =  V2. 

Pages  97-98 

I.   400  sq.  in.               3.   7.5,  7.5.  4.  -V2. 

5.  78V8sq.  in.             6.   20  V3/3,  20  V6/3.  7.  14  X  l^X  28  cu.  in. 

8.  48  mi.                      9.   44^,  approx.  10.  x=4  Vl5,  L=8  Vl5. 

12.  radius  =  vertical  side  =  4.20  ft.,  approx. 

Page  100 

4.  .3,  10/9, .,  .,  6, 14-, -,  ^9,  %.  5.   1.6,145.5. 
>/.   X  =  5.                           S.   y  =  40/x\                10.    -2M 

Pages  102-103 

I.    (a)  -4M -,-,-,  180/0:41^  etc.  2.    -.411. 

3.    -.016.  4.    -3.  5.    -1.28. 

6.  -2.5.  7.   $240.  8.- 7.83*  ft.,  6.52*  ft. 

9.  9.28*  ft.,  4.64*  ft.        10.  r  =  5in.,;i  =  5in.  12.   r  =  3.97* in., 

_  _  h  =  15.87*  in. 

13.  V20.      14.   40  in,         15.   66+6V40sq.in.  16.   65.98*  sq.  in. 

17.  r  =  /i=  V231/7r  in.       18.   270  sq.  in.  19.  600  sq.  in. 

Page  106 
I.    . . . ,  550/X12,  . . . ,  _6  /.  3.   12096  x3-5040M 

*  Approximately. 


ANSWERS   TO   EXERCISES  v 

4.  80-32^,  48ft./sec.,  -32ft./sec.2.    6.    12,12.     7.    .008,  .004. 
10.   20  ft./sec,  8  ft./sec.2.  12.   0,  1600  ft./min. 

Page  109 
2.   t  =  12.  4.   18,  6.  5.  20;  8,800,000  units/min. 

7.    ...,-15. 

Pafiie  112 
I.   6x(m2-2),  etc.         3.   36x2^3^  etc.  4.  24cu.  in./hr. 

5.  (a) 400 x3(x4+25)99;(d)  -4 7r(l  -a:)2,(e) -420 ty{t^-l)Mg) 3/4 (8-0-. 

6.  incr.  1.058*.  7.    16.2. 

Pa^es  115-116 

1.  -,-,{-mSMdx/dt,   ...,  (l+168/L3)dLM. 

2.  32  TT  cu.  ft./min.         4.    .5  in./min.  5.    .7957*  in./hr. 

6.  .00159*  ft./min.  8.    .000707*  in./hr.      9.    .00955*  in.,  2.4  sq.  in. 
10.    -.3  units/min.         11.    .0087*  cc./min.     12.    .02188*  ft. /deg. 

Pages  118-119 
2.   .225  ft./sec.  3.   64  ft./sec.  4.   7.5  ft./min. 

5.  -40  V5  ft./sec;  t  =  30,  1000  ft.  6.    -92  ft./sec. 

7.  %VllO.  8.   21.69*  ft./sec.  9.   32.49*  ft./sec. 
10.   incr.,  14  mi./hr.          11.    1.44;  96  mi.            12.  .533*  in./min. 


2. 


9 


_      Posfel21  _ 

4  Vx/3,  . . . ,    -5/2  Vx^  3.    .  • . ,  -40/3  Vx^. 

4.   .0053  sec.  5.   .037  yr.  6.  756.    

7.   .122  mi.  8.   .4  ft./sec.  per  ft.       9.    (a)  7  x^  ^^+1)*. 

(f)  8(x-5)/  V(x2-10x+7)3,  {i)  400 <V9 ^(25-1^)^     10.  3  hr. 

Po^e  123 
I.  dy  =^2  xdx,  etc.  4.  dx/df,  etc. 

Pages  124-125 
3.    ...,  -20/x5  -28x/(10-a;2)3.     "    4.   $12.48,  2.88,  7.21  ft.,  approx. 
S.   110  yd.  each.  6.    144  tt  cu.  in./hr.  7.  2.4  sq.  in./min. 

8.  24cu.  in.  9.    .0318  in.  10.   12;  4. 

12.   5;  300.  13.    . . . ,  30  V2  ft./sec.      14.   45i%3  im./hv, 

15.  i'^%6o.  16.    -500  per  in.;  625;  -^^728  in.perft. 

17.    ...,i  =  100;  -1600 ft./sec. 

*  Appro2dmately. 


VI 


MATHEMATICAL   ANALYSIS 


Pages  130-131 

I.  (b)...,  S^'i-2x+C,     (c)...,  ¥2x'-\-kx+C. 

3.    . . . ;  2/=%  V^+25  x-%  x4+25%.  5.  671  lb. 

8.   10,000.  9.    1100,  8400,  etc.        n.  83.3  lb.,  approx. 

Pages  133-134 
2.   60+80^-16  «2;%+VlO.  4.    ...,  4200Oft.,50±20V5. 

^.-80  V30  ft. /sec.  6.   t  =12.22;  -441  ft./sec,  approx. 

8.   <  =  15;  450  in.         9.   468.75ft.  10.    -98.71  ft./sec. 

II.  1600  ft.  12.   45  ft./sec,  675  ft.       16.   2.67;  ysa;^. 


Page  137 

I.  150.       2.   114. 

3- 

19375.       4.  40,89/11, 

3%,  11.25,  .105,2. 

5.   1248.75. 

6. 

32.              7.    256/3  ft,..ib. 

Pages  140-141 

I.   7741b.-ft. 

2.   7500,  380,  -,  328. 

3.   250  in.-lb. 

4.   30000  in.-lb. 

5.   2  dyne-cm. 

6.   3933^3  mi.-lb. 

7.   1.4(10-20)  dyne-cm. 

8.   992  TT. 

9.   486eu.  ft. 

16.   972  cu.  in. 

II.   .3  X  cu.  ft. 

12.   533V3CU.  in. 

14.   5236  cu.  in. 

- 

15.   5333y3cu.  in. 
Page  145 

4.   20000  TT. 

5.   309  X. 

6.    ...;  3485.1  cu. 

7.   11670  cu.  ft. 

8.    11666%  cu.  ft. 

9.  66%cu.  in. 

10.   %hr^. 

11.    144  cu.  in. 

12.   5333^/^  cu.  in. 

13.  85V3CU.  in. 

14.   2666%  cu.  in. 
Pages  149-150 

I.   8437501b.         3. 

1,406,250  1b.       4.   9375001b.       5.   3751b. 

in. 


6.    (c)-(e)    -V3(25 -x2)t+C,  i%Vx5+9+C,   -VssiS-j^y^+C. 
8.   2666%  lb.  9.   5208V3lb.  11.   Final  value,  36000  lb. 


Pages  152-153 
I.   .000  000  000  693+.     2.   .000  000  01354+.     3.   51b. 
5.   50000  1b.  6.   .4.  7.    1333y3cu.  in. 


4.  22501b. 
8.  60001b. 


II.   72  cu.  in.  13.   6000  in.-lb. 

15.    1000  ft.;  approaching,  128 ft. /sec.    16.   «  =  Vl25. 

17.    12.5(1+ V5);  12.5.  19.   254.68.         20.   61.42  sq.  in. 


ANSWERS   TO   EXERCISES 


vn 


Page  159 

4. 

607  ft. 

5.   614  ft. 

6.   2101b.,  22^ 

7. 

700  lb.,  38°. 

8.   200  lb.,  37°. 

9.   550  lb.,  53°. 

10. 

429  lb.,  473  lb. 

II.    115  lb.,  231  lb. 

12.    1000  lb.,  866  lb. 

13. 

13100  lb.,  10700  lb.  14.    311  tons,  238  tons. 

15.    180000  lb.  156000  lb. 

4- 

5     V63 
13'     63' 

Pages  16&-166 
5.    .09,5°. 

6.   548  ft. 

7. 

9". 

8.   63.7  lb.,  186  lb. 

9.   22°. 

10. 

1410  ft. 

II.    1.38  mi. 

12.    12.3  mi.,  49.5  mi. 

13- 

62°  N.  of  E.     14 

.  5.2%,  etc.     16.   5°,  etc. 

17.   76°. 

18. 

22°. 

19.    14400  lb.,  etc. 

20.   10.5  ft. 

21. 

4830  sq.  cm. 

22.   23°,  etc. 

2S.   391  lb.,  etc. 

26. 

60°. 

28.   31°,  etc. 

29.   4.79  mi./hr.,  etc. 

30. 

24.7  ft./sec. 

31.   58°  N.  of  E.,  etc. 
Page  168 

33.   24000  ft. 

2. 

944  ft.,  etc. 

3.   31°,  etc. 

6.   .918,  etc. 

7. 

8506  ft. 

8.   2250  1b.,  etc. 

9.    123  in.,  1170  sq.  in. 

10. 

1580  mi. 

II.   3630  mi. 
Pages  170-171 

15.    17900  cu.  in. 

I. 

9.21  ft.,  etc. 

2.   30.3  in. 

3.   6.05  in. 

4. 

107  sq.  in. 

6.    122  sq.  in.,  etc. 

7.   559  cu.  in. 

8. 

205  sq.  in. 

9.   308  lb.,  948  lb. 

II.   337  1b. 

12. 

N.  5.37  mi./hr., 

etc.    13.   S.  15.8  mi./hr. 

14.   13.8,  10.8,  ft./sec. 

15. 

3.03  m./s.,  etc. 

16.   9481b. 

17.   95100  ft.-lb. 

18. 

.866  sq.  ft.,  .342 

sq.  ft. 

Pages  173-174 

I. 

350  1b.        3.    185001b.,  etc.         4.   30001b 

.,  etc.        6.  6531b.,  etc. 

7. 

1331b.         8.   65001b.        9-   43600  lb.,  35700  lb.       10.   2301b. 

Page  175 

2. 

.08985,  . . . ,  1.0735.                             3-   955.36  ft.,  2°.87. 

5- 

111.07  sq.  in. 

6.   628.32  sq.  in.,  362.76  sq.  in. 

7- 

1.0515  in. 

9.   311.761b.  [w)  =  6 -2.6864  x]. 

TO. 

1.6782  ft. /min. 

II.    .00187  in. /min. 

Page  177 
(I)  79°22',  213.22,  695.68.  (II)  623.01,  71°1'.5,  55°41'.5. 

(Ill)  69°53'.4,  72°0M,  392.97;   or,  110°6'.6,  31°46'.9,  217.62. 


viii  MATHEMATICAL   ANALYSIS 

Pages  179-180 
2.   439.20,  315.68,  etc.  3.   482.77,  53n'.4,  etc. 

5.   5115.8  yd.  6.   3177.5  yd. 

7.  91,570,000  mi.  8.   701b.,  38°12'.8  from  2nd. 
9.  Incr.  58.06  ft. /sec. 

Page  183 
I.    ...,  .77875,  .62733.        3-    13  tons.  4-   75°31'.4. 

5.    108  mi.  6.    1014  ft.  7.   683.78  ft. 

8.  (i)  99°28'.4;  41°4'.8,  etc.  (iii)  4.493;  97°1',  etc.  (v)  9784.5; 
34°57'.5;  etc.  (vi)  76.880;  49°2r.9,  etc.;  or  26.212;  130°38M,  etc. 
(vii)  11189;  5023.1,  etc. 

9.  (i)  111240.     (iii)  14.517.     (v)  19,188,000.     (vi)  1134.7;    386.89. 
(vii)  24,609,000. 

Page  184 
I.    1821.1ft.  2.   1809.5  ft.  3.   2867.5  ft. 


Pages  186-188 

4.   47°11'.4. 

5.    12078  yd.,  6°27'.9  N.  of  E 

7.  743.37  sq.  ft.,  ( 

or  346.63  sq.  ft.,  etc. 

9 

126.42  ft. 

II.  82*'5'.8. 

12.  223,200,000  mi. 

13. 

21.048  ch. 

16.    1683.9  cu.  ft. 

17.    10  ft. 

19. 

8.16  in. 

20.   4  in.,  3.3282  in 

.  21.   9234;  11877. 

23- 

204.021b.,  22°0'.4. 

24.   2865.3  ft. 

25.    3104.9  ft. 

26. 

182.49  ft. 

27.   224.45  ft. 

28.   73°18'. 

29. 

3.775  X  10^5  mi. 

30.    10r32'.2. 

31.   .375ft./sec. 

32. 

233.4  cu.  in. 

33.   32235  1b. 

34.    ...,50°ir.7. 

35- 

17°39'.4. 

36.    .0031215  ft./min. 

37. 

.2  Ib./min. 

Pages  191-192 

3.   2.73  X  109. 

4.   50.9.                  6. 
Page  198  * 

4670  yr. 

9.  9.44. 

4.    (a)-(e)      3432; 

965,200;    7,260,000; 

2.438; 

979.9. 

ifUj)    7.235; 

1.796;           3.347; 

26.22; 

10,210,000. 

7.   41.52;  .00191. 

9. 

3.78. 

Pof^es  200-201  * 

2.  (a)'ic)    1.288;  .04756;  .0004569.  {d)-{f)    .001947;  .0007791;  .002025. 
(g)-(i)    1.262;. 2892;  .001913.  (j)-(O    4.168X10-6;  .01545;  .0002235. 

3.  .02129;  13.81;  7.706;  .7171.       4-   2.015X10^2.     5    io.91. 

*  The  answers  given  in  groups  are  not  always  in  order. 


ANSWERS   TO   EXERCISES  ix 

Page  202  * 
3.   -.-211,  .---989.  4.   -.-093  sec.  5-   S20530. 

6.  $3768.80.  7.    .--822  ft.  8.    10.552. 

Page  206 

3.  {o,y{c)     --.403,    .---868,     .--902. 
(d)-(/)      .---8872,    -.-068,     --.212. 

4.  -.-344.  5.  -.-988X1011.  6.   2361.5  cu.  ft.,  etc. 

Pages  208-209 
I.   --4.15;   -.-390.         3-    -11.938.  4.    134,220  sq.  cm. 

5.  384.86  ft.  6.    19°44'.  7.    (±?)  .16057,  etc. 

Page  212  f 
I.   $10,421,000,000.  2.   $3979.60.  3.   5.7%. 

7.  4.64'o.  9-   $1104.20.  11.   C3359.10. 

Page  214 
I.  2?3219;  2.1827,  etc.     2.   9.45+,  28.08,  13.13,  etc.      3.   $30072. 

Page  217 
3.   42''27'.7;  54.707  in.;  etc.    5.   982.92;  816.70;  etc.     6.   980.57. 

7.  (i)-(iii)    415.42,  169.65,  etc.;  .67640,  .70432,  etc.;  11.007,  20.497,  etc. 
(iv)-(v)    1134.2,  999.9;  etc.;  46.289,  4.2878,  etc. 

8.  44.088  ft.  9.  20°21'.9. 

Page  220 
I.   27'*31'.2,  91°18',  6in0'.9.  3.    (i)  38°41'.2,  86°37',  etc. 

3.  (iii)  113°4'.2,  36°56M,  etc.  (v)  64°41'.8,  33°47'.8,  etc. 

4.  S0°31'.3.  .        5.   88°42'. 

Page  222 
I.   38°14'.9,  ...,  70.233.   2.    ...,  48°31 '.7,  50.938.    3-   547.84;  929.72. 
4.    (i)  . . .,  110°45'.4,  1944.4;  1,169,000.  (iv). . .,  38°14'.3,  7.8946;  10.172. 
(vi)  . . . ,  40°3'.5,  167.40;  5186.6. 

Pages  226-227 

1.  (c)-(f7)      421.33;  -.13764;     54.708;     1.0931;     -.25546. 
{h)'{l)      -7.1470;    .0012565;   9965.5;    2.4304;     626.69. 

2.  $5306.70.  3.   $3494.20.  4.   4.52%.  5.   22.5. 

*  Ea«h  dash  stands  for  a  figure,  perhaps  zero,  in  the  answer. 

t  These  answers  were  obtained  from  five-place  tables  and  are  consequently  somewhat  inac- 


MATHEMATICAL   ANALYSIS 


6.  .19708.  7.   88.191ft.  8.   369.17  ft.  9.   173.13  ft. 
10.   389.79  ft.    II.   154.03.       12.   380.76  ft.  or  452  ft.    13.   1650.7  ft. 

Review  Set,  pages  228-235 
*  Chapter  I 

2.    -4^  sq.  ft. /ft.;  160  cu.  ft. 
3.    (19=b  V249)-T-14;  -2.33,  .20+,  2.13. 

Chapter  III 

7.  $792.35.  8.   «  =  4;  80.5  cu.  ft. 
12.    14417  units /min.;  .14716  units/min. 


II.   X  =  2,  Ax  =  .005. 


2.    ...,42000ft.; 
5.   218,750  1b. 


Chapter  IV 

.,  1639.52  ft. /sec.       3. 
7.    9  TT  CU.  in. 


50.4. 


24°!' 


Chapter  V 
4.   13584  lb.,  18376  lb.        6.   297.6  ft. 


4.   37500. 
8.   1.08  TT  lb. 


7.   102331b. 


Chapter  VI 
I.   73.42;  1.5416;  (=«=?)  .17394.  2.   26,017,000,000  mi. 

3.   13.51.  4.   99.53  ft.  5.    .35158  mi. 

Miscellaneous,  pages  232-235 
i.    141.03  ft.         2.    .01749  ft. /min.    3.   50000  ft.;  <  =6;  10935  ft./min. 


f2^00  7rxdx.    6.    27000  tt  cu.  ft. 

Bytrial,  «  =  13-.       11.    1875. 

-8.2571  Ib./in.,  1330.9  in'.-lb. 
18.   45°;  etc.       20.   4.1669  cu.  in.;  etc. 
23.   300  Ib./sec.  (Constant). 

189,5901b.  27.   78.446. 


5. 

8. 

14. 


26 


7. 

2000  ft.,  -208  ft. /sec. 

12. 

2.87+. 

16. 

8.75%.*       17.   21.6-.* 

21. 

2098.1  ft.      22.   98.33  lb, 

24. 

$2636.20.  25.  85in.-lb. 

28. 

118,150  cu.  in.,  approx. 

29.    1.869  ft./sec.         30.    .0129. 

Pages  240-241 
I.   $2473.10.        2.   5.13%.  3.   t  =  24.08.       4-   $41161. 

5.   5.824%.         6.   91492;  203610;....  7.   448. 

8.   t  =  2.31.        9.   68.387  mg.       10.   82.088,  t  =  9.21. 
II.   9.902.  12.    3.922%. 

Page  243 
I.    10008686<^  _  _  2.    . .  .,  12.5353,  7.3132-10,  3.0088-10. 

♦  Substituting  x  for  an  expression  gives  a  cubic  equation. 


ANSWERS   TO   EXERCISES 


3.    1.8379,  . . . ,  4.7574-10.  4-  -,  202,  -,  .457,  .0454. 

5.    1.597,  ...,.002813,  .1247.  6.  2.305,7652,6.419. 

7.   .09119;  10.9-.  8.  e-esau     g.2332»^ 

3.   t  =  4.48-. 


I.  /  =  36/x2. 
S-   R  =  3000  e--' 
8.   p  =  2000/fI 


2.   .69330. 


Page  247  # 

5.    . . . ,  Q  =  50  e-50^.  6.  21.972%. 

Pa^es  254-255 

2.   One  C.  /.  L.,  one  Power  L.  3.   D  =  29.4  e"  "i. 

6.    F  =  .25e0693ir  7.   AT  =  100  e-!^*. 

9.  r  =  Vz>3. 


.^x     3      -28 


Pa^es  259-260 
7.    (aHc)  —,...,  ^^;  -, 

X  XX 

8.   2.3026. 


11.  5.056.  12.    +260.6  ft. 

,,,   .86858  X        -2. 17145  .r^ 
14.    (b) 

16 


a:2-25  l-x' 

l-x2         2(1 -x^)       2a;2-l 


3    -5      1 


10.   486.6. 

13.   465.3  ft. /min. 
4 

15-    ...;-. 


2a; 


x{x^+l)'    x{x*+l)'    x(x2-l)'    (x2-l)(a:2+l)* 
2a;»  1  ^  1  dy 


x^-\-l     jjlogx 
a:2+l 


y  dx 


Page  2Q1 
;...;  14a;(7x3-l)(a:3-i)3;  5x2(5x4+3)/ V^^+l. 

Pages  263-264 
. ;   -56  x/e*-^;  Q(e^'  -e-^').      2.    (a)  2.1972  x  3*\ 


(x2-l)2 


I.   1.2e-o«*;  .. 

9.   i  =  30  e-«o^     13.   995°.*     15.   23.6  in.    17.   n  =  ATeK 

Pages  266-267 

^_^^^_2x(3x»+4.^+io)^  (c)  4xlogx;  (d)  2xe-(2:.+l); 

(x^— 5)2 

2x(25-2x^)           l-61ogx  -2x(x^+l) 

C/)     — , ;  (^)  :; — ;  (h) 


V25-X* 


X'  (X4-1)2        ' 

*  By  five-place  tables. 


xii  MATHEMATICAL  ANALYSIS 

3    ,-  1 

2.  (a)  X  =  V«,  ?/  =  —   (max.).     (6)  x  =  %,  n  =  ...  (max.). 
o  e 

(d)  min.  at  —  >  50,  max.  at  VoO. 

4.   30  cu.  in. /min.  5.   inc.  50  ttcu.  in./min.   6.    .2424  deg. /min. 

7.   dec.  2.1875  per  min.  8.    dec.  .1  per  rain.        9.    inc.  .00170  per  min. 

II.  S  =  1/2  e,  when  x  =  Ve.  12.   x  =  2  V2. 

Pages  268-270 

2.  10/x,     10(logx)Va:,     (1-log  a-)/x2,    2  xe^^     3e'{e'+iy. 

3.  (a)-(c)  1.30287 (logio  x^/x,     (3  x+4)/2  .r(.r  +  l),  .  (1  -5  log  x)/x«. 

2x(2x-'  +  l)  '—4 

(e),  (/),  (/i).        \-—     ;    2x3e-io-(2-5x); 


(i),  0').  2x(loga:)2;      2x^62^ 

5.  X  =  ±V2/2.  6.    inc.  4.375  per  min.      7.   x  =  200(1 -e'^^. 

8.  8.664%.  9.    13.86.  10.    32.894. 

12.  Only  .3%.  16.    /  =  600/x2.  ^  17.   /  =  100  e'-^. 

18.  5  =  6.2e-'i5«,  approx.  19.  /  =  18400/d*,  approx. 

20.  JS  =  29.2e-*626f;   Hess. 

Po^es  273-274 
2.   FC  =  4.919mi.,26°34'N.of  E.    3-   13892  m.,  30^5 '.4  N.  of  E. 
5.    . . . ,  49.729  mi.    6.    18.4  ft.  7-    422  sq.  ft.         &    814  sq.  ft. 

10.  1494  ft.  II.   t  =  5;  X  =  1200. 

Pages  276-277 
2.   913ft./sec.,28°49'up;etc.    3-   ^^  =  30000;  v  =  1000;  7  = -36'*52'. 
5.   3+3^2.  76.  8.   2(17^-1);  24;  224 

Pages  278-279 
I.   X  =  1399.4  f,  7j  =  540  t-mtK       2.   t  =  33.75,  x  =  47230,  . . . 
3.   f  =1%6;  (135.32,  39.06).  4-    •••;  580.9ft. 

Pages  281-282 

I.   10,...,  201.91.      5-    -54°28',^.  11.  750. 

,12.   V2873,  V3793.     14.    VIoO,  Vl57,  etc.  15.  Yes,  no,  yes. 

18.   31°16'.  19.   74°30'  (if  prolonged).  20.  16.35  mi. 


ANSWERS   TO   EXERCISES  xiii 

Pages  285-286 
2.   No.  (Test?)  8.    (4,  4);  also 12.   The  inscribed  circle 

13.  a;*+2/'  =  100.  14-   x'^-y'  =  16.  15.   x'^+y''  =  64. 

Page  288 

7.  No;  no.  (Test?)  10.   Unlimited  line,  or  sides  of  a  square,  accord- 

5.  2  x—y  =  9.  ing  to  interpretation  of  "  distance." 

Pages  290-291 
2,    (6)(j--5)2  +  (2/-4)2=25.        (c)  (x+2)2  +  (?/-3)2  =  37. 
4.    (-8,  15),  r  =  17;  etc.  5.    ...;(-%,  0),  r  =  %. 

8.    (.r +4)2 +7/ =64.  9.    (1.5,7). 

II.    ...,16;  etc.  13.   200;  150;  . . . ;  32. 

14.  ...;  22  77,  lOir. 

Page  293 

6.  Straight  line.  7.   Circle,  or  straight  line. 

8.  Circle.  (What  center?)  9.  y^  =  12  x,  simplified. 

Pa^es  296-297 
2.   F  (0, 1/4),  Dir.,  y  =  -M]  etc.  3.   If  in  doubt,  read  §  198. 

5.   x2  =  25  ?/.  6.    . . . ;  2.  7-    •  •  • ;  (ie.,  y  =  .000142  a:^);  etc. 

8.   X'-  =  1600?/; etc.     9.   6ft.,  etc.      13.   x^  =  lOOy',%  in. 


Pages  298-299 

I. 

\0,  -10),  etc.            4.   a:2  =-4  2/. 

5.   x"^  =  -400?/;  3  in. 

6. 

7.4  in.;  -.0463.        8.   24.8  ft.,  63.2  ft. 
Pages  302-303 

.t2                ?/2 

'^     400       144 

z. 

(±15,0);...,(0,  =^2V5).    4.    (c)§  + 

l-^<<o+i£- 

7. 
10. 

b  =  92,986,000;^etc.           .   8.    (a)  38.4; 
5.508ft.;etc.                  11.    ^+|  = 

Pages  305-306 

(6)  40,  when  re  =  5/  V2. 
x"^          y- 
'■        "■    405  +  144=^- 

I. 

500  IT  sq.  in.     4.   250p  tt  cu.  in.        6.   8  ft.;  etc.       9.  %  =  cos  C; 
^  -  SeO^^''^-     "■    2500  +900   =  '•     lU,e  similar  triangles.) 

13- 

400      225 

xiv  MATHEMATICAL   ANALYSIS 

Page  309 

I.    . . .,  c  =  5,  VlS^  2V2.  2.   c  =  10,  a  =  6,  /.  6=8;  etc. 

4.   Two  parabolas,  and  . . .?  5.   Two  hyperbolas  (4  branchies). 

6.   Two  parabolas.  (Why?)  7.    -^^  -  ■—:  =  1. 

400       22o 

Page  311 

1.  (0,  =t3  Vu);  etc.  2.   G(6780,  5480).    3.  G(7970,  6930).  4.  ^V  =  50. 

Page  314 
3.   3  parabolip^  4  hyper.;  1  straight.  4.   xy  =  V2k\  7.  69.315. 

Page  317 

2.    (a),  (6)'.     Foci:  (18,  -6),  (-2,  -6);  (-10,  ±5). 

(y-6)^  (x+l)2        ,     .     .   ,     ,    ,w     1    11^ 

(c)       ,^ —   =  1;  foci,  (-1,1),  (-1,11). 

16  9 

fx— 4)2         (v— 4)2 
4.  -^TT^  +  -^1^   =  1.       5.   Vertex,  (8,  1);  p  =  3;  etc. 

lb  25 

(x-6y       (2/ -2)2 
6.  -— -  +  ~~-  =  ,1.      8.  {x - 15000)2  =  -22500  (1/ - 10000). 

25  16  '^ 

10.    (25-48000)2=:  -4802(7/ -10000) ; i5'(48000, -47600);  Dir.,y=Q7,mO. 

12.   x2(l-e2)_20x+?/2  +  100=0;     ...;    hyperbola. 

Page  321 
I-    (d)'{f),ih)    a:-62/+48=0,6x-7/  =  41,x+?/  =  7,3x-2i/+25=0. 

2.  2x-?/  =  16,3x+7  2/  =  109,etc.  3.  3  J-IO  ?/+ 61  =  0, 

15x+2/  =  103,  etc. 
6.    (V4,  2),  4x-2/+l=0. 

Pages  322-323 
I.    (7%i,  23Ai).  3.   2x+i,'  =  15,2x-5!/+19=0,  (^%,i%). 

4.   Meet  at  (%,  %).  5.    (%,  -^^Z^). 

6.  (2;-2)2  +  (?/+l)2  =  100.  7.   Center  (^^.-i,^  29/23)^ 
8.    (,-1,1)-  9-    (15,30). 

10.    (5;  10).  II.    lOx+j/  =  19. 

Pages  324-325 
4.    16X-2/2+18  7/-65  =0.       6.    16x2+257/2+192x-200?/+576=0. 

7.  32  x-y^-25Q  =  0.  8.    (x +9)2/225 -z/2/400  =  1. 


ANSWERS   TO    EXERCISES  xv 

9.  Same  as  in  Ex.  8.       12.  (3,  6),  (0,  6). 

13.  9x'-my^-lUx-\-l2Sy-\-176  =  0. 

14.  x2- 120000  a: +  1600001/  =0;  (60000,  -17500). 

15.  y=   -.00092098 a;2.        16.  3.8%. 

17.   About  48°  S.  of  W.  i8.   y  =  V20  x^+C. 

19.    Y'-  =  SX.  21.   Hyperbola  (Why?);  85.836  in.-lb. 

22.   P  =  .05  F2;  1. 

Page  328 

3.    =*=3,  ±1;     3,%(-l±  V^);etc.     4.    13.49.        5.    10.39. 
6.   .5193  I  7.  E,  P,  //,  pair  of  lines. 

Pages  329-330 
2.    y,  not  X  (Why?)        3.    (a),  (6)  Yes.    4.   k=%;  x=%,  ¥3. 
5.    (-4; -2, -2);  etc.     6.    -%,  etc.         7.   A:  =  50,  or  ..  .(Why  two?). 

Pages  333-334* 

1.  (a)-(e)    2,3,(-7±  V37)/2;     1,-4,...;    -2,-4,...; 
-1,-2,5,.^  1,-1,  ...;     (/)-(i)    5, -4,-1,...;  1/2,-3,  ...; 

%,-%,  =^V-i;    ^^,%,  ...;     -^^, -%,.... 

2.  (a)-(c)     (2x-l)(3a;+l)  (6x2+j;+l);     (2  a:-5)  . . . ;  (x+1)  . . .  ; 
(d)-(/)     (2a:-l)(3a;+l)  ...;  (2x+l)(2x-3)  ...;  {x-iy  (x^+l). 

3.  (8x3_^g^2  +  i)  ._.  4.    ...^  (2x-3+9x2-4). 

5.    (5+5V33)-^8.  6.    10.  7.   «  =  5. 

Pfli/es  338-339  t  '  ' 

2.   4. --6838.      3-. --33,    -4. --01.      4-  2. -- 84,  .  - -16,  -2.  - -00. 

6.   -.-34579.       8.    .6180,  -1.6180.        10.  .22.         11.    .66. 

12.   X  =  2.65.  13.   X  =  5.23.  14.   2.480,  2.510. 

15.   47.82,  .09 --8. 

Page  341  (top) 
3.   One  step,  from  2.3:  2.3099.         ■  4.   Three  real. 

Pages  341-342 
I.   2.166013.  3-   2.92.  4.   V2,  1.10;  -V2,  -2,  -.62,  .73,  4.40. 

5.  (x  +  l)...(4x2+8x+5);     (.u-1)  (2j:+5)  ...(4x2-2^  +  1); 
(x-iy  ...•  (.r2+4.c  +  l). 

6.  (3,  2),  (3.6,  -1.8),  (-2.8,  3.2),  ...  8.   32.23. 

*  The  answers  given  in  groups  are  not  in  order.  '"" 

t  Here  long  dashes  denote  minus  signs;  short  dashes,  missing  figures. 


xvi  MATHEMATICAL   ANALYSIS 

9.   6.4%.     (One  step  from  a:  =  1.03.)  11.    .6931;  1.6094. 

12.    VlO.                    13.   3.059.  14.   6.179.         15.    1.843. 
Pages  344-345 

1.  47.03  mi.              3-    • . . ;  2.64,  etc.  4.    15.8.                5.   30. 

Page  347 

2.  139.6  in.  4.   50;  187.5;  5890.4  ft.  5.   471.24. 

Pages  348-349 

3.  20in./min.;  4m./mm.2       4.   10.8in./sec.        5.    .54  iu./sec.^ 
6.   At«=30,81ft.  12.   .10472.  13.   7.27X10-6. 

14.   .288mi./sec.  15.   .202  mi. /sec. 

Page  350 
I.   17900mi.       2.   116  mi.         3.   4.64X108mi.       4.  2.51X10i3mi. 

5.  0".81.  6.   19.2  mi.        8.   9     mi.,  approx.  9-   Vao^"). 

Pages  358-359 

6.  ....  117°29'.2;   115^24',  . . . ;  . . .,  199°24'.8. 

8.  -.83898;  -.54416;  1.5417. 

9.  3.8877<''>,  5.5371(^^  .7671^^),  5.5161<^^  3.0787('>,  6.2203<'>. 

Page  360 
I.  0,  .0628,  .1257,  etc. ;  .0314,  .0942,  etc.   3.  Max.  at  t  =  .00196;  ~-  sec. 

Page  363 

5.  (6)  -4.8  sin  8 1,  (e)  10  tt  cos  f  ^  n ,    (/)  - .21  «=  sin  {i^). 

6.  (o)  5.2359  cos  (3  < +20).       7.   .000964.  8.   168;  54000. 

10.  V—  —1.40  cm. /sec.  11.   <•=  — 8in./sec.     12.  w  =  .6283('"V«ec. 

Pages  364-365 
I.  1.05(^),  . . . ;  l-e^*"),  ...;  %  TT  (»•>,.. .    2.  44.73. 
3.  .8097,  1.4903.  4-    •  • . ,  -105  cos2(5  0  sin  (5  0- 

Pa^e  365 

7.  273perrad.  8.    -.00597,  .0267. 

Page  367 
.3.    . . . ;  (-18.794,  -6.840).         4.    •  •  • ;  (25,  323^*8'),  or  (-25,143°8').      . 

8.  (%,  -Vs,  -%,  ...)  or  (-%,  %,  -%,  . . .).  9.  3530  mi. 


0)ctn0).    (/)-2cs.g)ctng). 


ANSWERS   TO   EXERCISES  xvii 

10.  6000  in./mm.;  300  in./mm.2         12.    ...;  -.389. 
13.  73.89;  2.5.  14.   1.0027(''>. 

Pages  371-372 
2.   cos  d,  —  sin  ^,  . . . ,  cos  d  —  smd,  cos  d.       4.    (a)  [Factor]. 

4.  (6)-(c)   sec0+tan0;  sec0.  (/)  —  2sin0. 

5.  d+C;  2  0+C.  6.  sec2  0;  -csc9ctn0. 

7.  (a)  53°8',  306^52'.         (6)  45°,  225°,  7r34',  etc. 
(c)  8°43'.7,  81°16'.3;  and  two  in  Q  III. 

8.  (a)  30°;  also  II,  III,  IV.  (6)  120°,  240°.   9.  2°45'.3;  87°14'.7. 

10.  50000.        II.  8°31'.8.        12.  45°. 

13.  50°21'.4.  14.  14°2'.2[tan0=.25.] 

Pages  373-374 

1.  (C)  -10  t  CSC2  (5  «2).    {d)  -   CSC 

(g)  ctn  e.        (h)  3  ctn^  6.        (j)  e'  sec  t  (1  +tan  t). 

2.  secx;  — csc0ctn0;  —esc  6;  sec^^. 

3.  cos2  0— sin2^;  ...;  csc^e  —  csc^ctn^;  sec2  0  —  csc^d; 

4.  1.6.  5.    1;  -2. 

Pof^e  375 

2.  80  TT.     4.   320.     6.   10V2(l-cosf).     7.   x  =  10cos  (2«+0- 
9.  30  sin  <  cos  2;  15;  15. 

Page  378 

I.  x=50i.2t -sin  .2t),y=  . . . ;  etc.     2.   6  Vi  in./sec. ;  12  in./sec. 

3.  -2ft./sec.  ,...,  .3328ft./sec.2        4.    -44.82  in./sec. 

5.  a; =20  cos  (2«-x/3),  . . .,  -9.787  in. /sec. 

Pages  380-381 
I.    (a)  Min.  at  t  =  .0067,  max.  at  t  =  .0145.      2.    -.1  cos  10 1  +C, 
(c)  Max.  at  t  =  .2487,  min.  at  t  =  .7487.       3.   Q  =  .0159+. 

5.  (6)  (. . )  tanS  e+C,        (c)  (.  . )  sin-3  0+C. 

6.  (a)  log  sin  ^+C,  (6)  2^-2  cos  0+C. 

7.  (c)  -COS0+C,  (/)sin0+C, 
(^)sin&— COS0+C,      (/i)  — COS0+C. 

Po^es  384-385 

7.  4cos3  0-3cos9.  8.    (d)  (cos  A  +  V3  sin  A) /2. 

9.  ^%5,  ^%5.  10.    ^%5,    — ^%5. 

11.  (6)  tan  5  d,  (c)  cos  A.  12.  5  V3  sin  3  «+6  cos  3 1, 

14.  c  =  10,  A  =  36°52'.2  =  .6435^'^. 


xviii  MATHEMATICAL  ANALYSIS 

Pages  386-387 

1.  {b)V2d-VL2smQd+C;  etc.  2.   34'.3. 

6.  (tan  A  +tan  B)  -r-  (1  —tan  A  tan  B). 

7.  8°7'.8,  19°26'.4,  etc.  8.    (=»=  2,  4),  77°28'.3. 
9.    (6,  2),  52°7'.5;   (-3,  -4),  86°49'.2. 

Page  388 

2.  -ctn  4  6,  tan  6  5,  ctn  30°  tan  20°,  -tan  25°  tan  5°. 

3.  V2  (sin  10  0+  sin  2(9),  etc. 

Pa^e  389 

1.  (6)  y2  [xVa;2-25  -25  log  {x+^x^-25)].  Use  (41)  with  m  =  0;  etc 
(e)  %  tan5  d  -Vs  tan^  0+tan  0  -d-\-C. 

(/)  Vs  sin  0  sec2  d  (2  sec^  0  +3)  +%  log  (sec  0 +tan  d)  +C. 

(h)  V9J^{S\ogx-l)+C.  (i)  -VL3e-^\2 sin St+3co8 3 t)+C. 

2.  6  TT.     (Cf.  §  214.)  3.   7rV2. 

Pages  390-391 
2.  2^221,220/231^2^220,  etc.  5-   3  sin  0 -4  sin^  0.        6.   45°. 

7.  sin-M.005F).  8.    .141  deg./deg. 
9.  c  =  .97985;  75°49'.3.                    10.   38.114  ft. 

II.   45°N.  E.  12.   A/2.  13.   45°. 

14.   1.2113,  .2218,  41°42'.9.  16.   75°49'.3,  or  ...;  165°49'.3. 

Page  394 

I.  248;  97.2;  134.14;  1.    2.   2.  3-  2. 

4.  1318.3.                          5.   102,500.  6.  238  7r/3  cu.  ft 
7.   7r-7rV4.                        8.   750  7rcu.  in.  9.  50001b. 

Page  398 
5.   20%.  6.  Vs.  7-   180001b. 

8.  270000  Ib.-ft.  9.   4000/3  Ib.-ft.  10.   250  7r/3  lb. 

II.  8001b.  12.   752680  ft.-lb.         13.  3333V3. 

Pages  401-402 

1.  (a)  5%.  (c)  20/^+3  log  3.*  2.  (a)  (Vl000-l)7r/27. 

2.  (6)416  7r/3.        (d)48V5.               _            4.  (a)  18.33.* 
4.    (6)  k  [(  x2  +4)t  -8]  -7-  3.      (c)  V2  (e^-1).  (d)  10  tt. 

6.   2133y3  Ib.-ft.  7.    128000  TT  ft.-lb. 

*  Use  formula  (41),  p.  497,  with  m  =  0. 


ANSWERS   TO   EXERCISES 


XIX 


90. 
819.2. 


16^%i. 
(d)-3,  6. 
id)  2,  12. 


125%05. 

171%. 


Page  405 
3.   78. 
7. 


11.25. 


(e)  5,  4. 


Pafifes  407-408 

3.  (a)  4,  4.     (6)  -12,  - 

4.  (a)  2,  4.     (6)  2,  1. 

6.    (a)  (3,  -1,  17),  min. 


16. 


10%. 
170%. 


(c)  (0,  1, 17),  neither,      (d)  four  max.,  one  min. 
Page  411 
2.  y  = 


lOSVs. 

.0795  (=5%) 
170%;%;... 


70  cu.  ft. 
672  -If  cu.  in. 


1%). 


I.  10  Vs. 

4.  ^%. 

7.   16000  Ib.-ft. 


leVs. 
.0349  (  = 

40/7r. 

Pages  412^13 
5940  IT  cu.  in. 
1200  TT  cu.  in. 

Fa^e  414 

2.    12iy27. 

5.   360  tt;  720  tt. 

8.   800  7r/31b. 

2l 


10.  2400  7r2  Us  comes  out  48  ttV  100 

11.  76800  lb.  Iw  comes  out :  3  Vx]. 


2. 

5. 
8. 

13. 
16. 


900;  2500.  3- 

$1252.50.  6. 

...,  16(1.530-1),     10. 

$12080.  14. 

$285.75  (=$22.50+*S) 


Pages  417-418 
420;  7656. 
$4350. 
.003171  W. 

$642. 


$22607. 
$1188.90. 
Life,  by  $3.53. 
(6)  $8.47-. 
(e)  $21.47+. 


Pages  421-422  f 
s.   $8480.20. 
;.   $37934. 

;.   $8.84+. 

(c)  $16.06-. 
K   $187.66. 


(c)  5,  10. 

(c)  6,  4. 
(6)  (0, 0,  40),  max. 
7.  (6,5). 


2/=  80. 

17.26;  370.67;  .7072. 


1280  TT  cu.  in. 
304  7r/3  cu.  in. 


3.1  in. 
1.00003. 


12.  30%cu.ft. 


4.  31360;  99500. 

7.  $615. 

12.  3435.6a* 

15.  $1027.50. 
17.    V2,  V2. 


3.   $1639.20. 
6.   $1685.54. 
9.    (a)  $14.99-. 
(d)  $74.30+. 


*  Using  six-place  logarithmic  tables. 

t  Five-place  tables  will  not  give  some  of  these  results  very  closely. 


XX 


MATHEMATICAL   ANALYSIS 


Pages  424-425 

I.   $7494.03. 

3.   $3618.33.                  4-   $3419.16. 

5.   $1971.16. 

6.   $58.96.                      7.   $55.85-. 

8.   $6682.20. 

10.    (a)  $63.91.                   (6)  $56.03. 

10.    (c)  $234.96. 

II.   $6065+.                 12.   $672.97. 

Po^e  426 

I.   $372.43 +$564.81. 

3-    (a)  $853.               (6)  $512.81, 

3.    (c)  $4805.13. 

id)  $10127.30.      (e)  $34.25. 

4.   $173.23+. 

5.   $334.59. 

Pfl^e430 

x^      re*      a^ 

,    X2         X3     ,     X* 

etc.             4.  X+-+-+-. 

5.   .09967. 

6.  x-x3/3!+x6/5!. 

II.   .493108;  .049986. 

12.  a»+5a%+10a3x2,etc. 

13.  a"+na"-ix+p^^a»-2x2.... 

Pages  432-433 

^...M"- 

(sr)  1+X+X2,  etc. 

1      2x'      x«      4x« 
^  •^  9       27       27       243  ■ 

2.    -(x+xV2+x3/3,  etc.).                 3.    -.05129,  -.05033. 

4.   x+Vs  x4-  ^56  x^+i^eo  xi°. 


5.   .5032. 


Page  435 

I.    (a)  log  V^Y  =  l.5708J\^^,    4.7124  V^,  etc.        

(6)  log  (.5+.86603  V-l)  =  1.0472  V-1,    4.I888  V-1,  etc. 

Pages  437-438 

3.  y  =  10x+3x2;  y  =  -7+.2x2;  y  ==  -5  -2x+.lx2. 

y  =  ll+x2-V4x3;  y  =  2.47 +. 0515 x+. 00165x2  -.00005 x». 
A  =  1.4x2 -.1x3;  etc. 

4.  2/  =  125-.5x;  y  =  20x-x2. 


6.   1.003984. 


Page  439 
i/  =  28-5x+^x«. 


10.  $1022.50. 


ANSWERS   TO   EXERCISES 


XXI 


Pages  441-442 

I.   120;  15600.           2, 

,   5040.         3.   6720.          5.  10,0001^9500!. 

6.   64. 

7.   504.                          8.   20!  X28!-i-  8!. 

9.   8. 

10.   1,814,400.               11.   48. 

12.   64,48. 

Po^es  445-446 

*  I.   (a)-(d)  252;  5005; 

38760 ;  12650.     (Z)   47 !  -^  (36 !  6 !  5 !). 

(n)   8,778,000. 

3.   45.                           4.   895440. 

6.   288. 

8.   30240.                    9.   16800  X  181 -M3!. 

10.   3696. 

II.  2880X13!           12.   11760. 

13.   5040. 

14.   56.                        15.    ...,  220. 

Pages  449-450 

2        240 
^*    1001     1001' 

3.  .34-.                4.  .004917,  .1598. 

5.    (c)  .007128. 

6.  .0401.                7.  f^. 

1 

1                           36 

8. 

11050 

«^*N024-         '"•*'343;      "•«'>-°«l 

Page  451 

I.  %6,  .05427,  .2257. 

2.   .288.                        3.   .512. 

4.   .3501. 

5.  %58e.                       6.  .000404. 

7.   .0108. 

Pages  455-456 

I.   9%,  91%. 

2.    ...,  28m.  closer.                   4-  ^• 

5.    ...,98%. 

6.   .4724,  .3829.                             7.  5.529. 

Page  458 

I.  88/105;  7.53. 

2.   .0000462;  etc.                  $-   (fl)%^Vso. 

3.   (6)  .8,  -1,  approx. 

4.   W  =  35.33+. 04175  r. 

Page  459 

I.   7.6185  X  101°,  approx 

:.           2.   .278,  approx.         3.   .0225. 

4.   TF  =  6.9823  T+105.16.                                           5.   .00811. 

6.  y  =  3.95  X  +52;  x  = 

=  .0805  y  -2.82;  i  =  .318. 

Page  463 

2.  92+34i,i7A3+%3i, 

-,    13+z,   8+2i, -,  2-lli,  %6-%3i. 

3.    =h[(3^-^)/V2,-,(l  +  3^)/V2,l-2^,3-2^,(l-5^)/V2,4+^, 

(3-5i)/V2,  -,   (9-7i)/V2,  2+5t,  5-6i. 


xxii  MATHEMATICAL   ANALYSIS 

Page4QQ 

3.  (6)  2  V2  cis  225°.  (e)  8  cis  150°^  (9-)  3  cis  180**. 

4.  (6)  3.2140- 3.8302  i.    {e)-Q-Q^ISi.        (/)4  V2(-l+i). 

5.  cis280<';-;  7.2812  cis  208°!'. 

Pages  467-468 

1.  (d)  30  cis 220°  =  . . . .        (^)  250  cis  150°  =  . . ..       (h)  —  i. 

2.  -.75i.  3.  2.1426-2.55351. 

Page  469 

2.  (6)  2  cis  75°,  2  cis  165°,  etc.     (e)  3  cis  140°,  etc. 

3.  .80902 +. 58779  i;   -.30902+.95106  ^,  etc. 

Page  470 
I.    (5)  cis  22°30',  cis  67°30',  etc.        (c)  (  ^ls+i) /2,  (l  +  V3  i) /2,  i,  etc. 
(h)  i,  -  (  V3  +i)  /2,  etc.  (i)  cis  22°30',  cis  112°30',  etc. 

General  Review,  pages  475-484 
Chapters  I-VI 
I.  150.  3.   45000  lb.  4-    11984  lb.,  etc. 

5.    146.45  sq.  in. ;  . . . ;  277.02  cu.  in.     6.   320000;  . . . ;  13500. 
8.   $197.72.         9.    -3permin.         10.   2298  yd.         12.   3.2282. 
13.   62.5;  127.93  sec.  14.   2560  lb. 

15.   26.67  deg./sec2;  291600°. 

Chapter  Vn 

1.  2a-/(a;2+l);  2.17145/x;  --;  a:(l+21oga;);  eV(e*+l)2. 

2.  . . . ;  log  a:2o+C;  V2  {e^^  -e-^')  -2  x  +C. 

3.  1/5  e.  4-   22.14%.  5.    F=20e-o6«r.  8.  1/ =  7.44/Va:. 

Chapter  VIU 

.Vh,...59»2'.  4.    <-^  +  '-^^=1,x--20(.-2), 

«%  16 

5.    (x -38400)2  =  -102400  (?/- 14400);    .... 

8.    ...;  3.75ft.;  (0,^00/3);  etc.  9.    (a;-3)2+(?/-4)2  =  25. 

Chapter  IX 
I.    ±2.3142,  ±.55786.        3.   Two  linear,  one  quadratic. 

4.  -1.6751,  -.5392,  2.2143. 


ANSWERS   TO    EXERCISES  xxiii 


Chapter  X 

5. 

40in./sec.             6.    -5.878, -1525  per  sec;    ^  =  ^120,  etc. 

Chapter  XI 

I. 

COS0.                      2.    —%  eosQ  t+C;  ...;   — logcos0+C. 

3. 

-i/2[csc(9ctn0+log(csc0+ctn9)]+C;  ...;   - 1^2  cos  6  0-1.4  cos  2  0. 

4. 

10°33',  79°27',  190°33',  259°27'. 

5. 

56     33        56                               16     63         16 
^+^^  -^'   ^'  -^'^'"'  ^-^^   -i^'   ^'  -S'  — 

6. 

.1  VlO;  2y25.         7.   -2  sec. ;  8;  -239.03  units  per  sec. 

9 

10.63,  5792;  t  =  .003678;  127  times  per  sec. 

Chapter  XII 

3.  16.5  7r  M7-V8  7rloge(4+Vl7).     [Use  (44),  (41),  p.  497.]     4.    16. 

Chapter  Xin 
2.  $104,  nearly.         3.   $18.74+.         4.  $65.90.         5.   $4180,  nearly. 

Chapter  XIV 
I.   358800.  2.    15504.  3.   302,400. 

4.  1.71  X  10-«.  5-    -1285.  7.   36  meters  nearer. 

8.   y  =  .90lx.  9-   2/  =  33.41+.52a:,x  =  25.72+.61?/;i  =  .3172. 

Chapter  XV 
I.    . . . ;  -2%7  +  i%7  i.      2.   cis  18°,  i, . . . ;  -;  1.3797  cis  10°37'.6,  etc. 

Miscellaneous 

I.  F  =  9e-»«93ir.  50.9.  2.    (x+iy+y^  =  4. 

4.  .342.  5.   45°,  90°,  210°,  225°,  270°,  330°. 

6.  59°2';  239°2'.  9.    -2,V3,2.12.  10.  .2  in. /sec;  5 in. 

12.  (6)  62500  TT  lb.  13.    1,014,500  ft.-lb.  14.    (a)xyix+l)K 

14.  (6)-(d)  sec  d  tan  9]  sec^  x;  [x  sin  .t+ 2  (cos  a:  —  1)]  -^ a:^^ 

15.  (a)-(c)  -Vs  csc3  «+C;  Va  {x^-100)^+C;  .01  a:/  VlOO-x^+C. 

16.  2.1752;  .3466.  17.    (?/ -9)7225 -xV400  =  1. 
18.  5.8%.  19.   Ellipse;  ?/=±2VlO,  x==i=2. 
20.  ...;  .10256.       21.   +$6.08.          23.   79°41'.7or36°52'.2. 

24.   ^260.  25.    .26;  .94.         26.   ^  =  100  e-»o%  Q  =  .  1986 +. 

27.6.53ft.  29.   165.  30.  49.6;  (10.85, -16.56);.... 

31.   5  VlO,  left  and  up  63 °26';  hyperbola;  69.31.        34.   80  ft. 


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